Composition of hypersingular integral operators

Horváth, J. I.

doi:10.1080/00036817808839189

Cited by 26 publications

(15 citation statements)

References 8 publications

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“…For the sake of completeness we now state this result for the kernels given by (8) and (9). The proof is the same as the one given in the one-dimensional case.…”

Section: ])mentioning

confidence: 99%

“…Such weighted spaces appeared naturally in the study made by L. Schwartz ([14, p. 214]) of Newtonian potentials of distributions, as well as in his paper [15]. For other occurrences of these spaces, see for instance [8], [9], [11]- [13], [1], [2].…”

Section: The Integration Formulas As Integrals Of S -Convolutionsmentioning

confidence: 99%

“…This notion has been studied and applied by many authors (see for instance [16], [15], [8], [6], [9], [11]- [13], [1], [2]). In particular, R. Shiraishi introduced in [16] an equivalent definition, which is the one we will use here.…”

Section: F(t * S) = F(t ) · F(s)mentioning

confidence: 99%

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S'-convolvability with the Poisson kernel in the Euclidean case and the product domain case

Álvarez

Guzmán-Partida

Skórnik³

2003

Studia Math.

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Abstract. We obtain real-variable and complex-variable formulas for the integral of an integrable distribution in the n-dimensional case. These formulas involve specific versions of the Cauchy kernel and the Poisson kernel, namely, the Euclidean version and the product domain version. We interpret the real-variable formulas as integrals of Sconvolutions. We characterize those tempered distribution that are S -convolvable with the Poisson kernel in the Euclidean case and the product domain case. As an application of our results we prove that every integrable distribution on R n has a harmonic extension to the upper half-space R n+1 + .

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“…For the sake of completeness we now state this result for the kernels given by (8) and (9). The proof is the same as the one given in the one-dimensional case.…”

Section: ])mentioning

confidence: 99%

Section: The Integration Formulas As Integrals Of S -Convolutionsmentioning

confidence: 99%

See 1 more Smart Citation

S'-convolvability with the Poisson kernel in the Euclidean case and the product domain case

Álvarez

Guzmán-Partida

Skórnik³

2003

Studia Math.

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“…Weigthed spaces of distributions. We will need the following weighted space of integrable distributions, introduced in the Euclidean setting in [Ho1,Ho2,Or]. …”

Section: Using This Lemma Inductively Givesmentioning

confidence: 99%

Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups

Damek¹,

Dziubański²,

Jaming³

et al. 2009

MATH. SCAND.

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Abstract. In this paper, we characterize the class of distributions on an homogeneous Lie group N that can be extended via Poisson integration to a solvable one-dimensional extension S of N. To do so, we introducte the ß ′ -convolution on N and show that the set of distributions that are ß ′ -convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of L 1 -functions. Moreover, we show that the ß ′ -convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behaviour. Finally, we show that such distributions satisfy some global weak-L 1 estimates.

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“…The investigation of hypersingular integrals in the case of the standard Bessel potentials (i.e., isotropic ones) has been carried out by Stein [10] and Wheeden [15], in the case of anisotropic potential spaces with respect to a diagonal dilation matrix by Lizorkin [5]. For related work see [9,3,7,1]. Our methods of proof essentially consist in using Fourier multiplier techniques.…”

Section: Introductionmentioning

confidence: 99%